Advancing Cosmological Simulations of Fuzzy Dark Matter with Physics-Informed Neural Networks(PINNs)

Ashutosh K. Mishra1 (PhD Student) Advisor: Emma Tolley

27 August 2025


1 1Email: ashutosh.mishra@epfl.ch

CMB Power Spectrum


BM:DM = 1:5 !



ΛCDM Theoretical Fit:

Credit: ESA and the Planck Collaboration


Ωbh2 ≈ 0.024, Ωmh2 ≈ 0.14

2

Small Scale Challenges in CDM Model



Potential Problem: Absence of Baryonic Processes (Feedback, Formation) and/or Nature of DM!

3


Oman et.al. 2015

Baryonic Processes


Strongly model dependent e.g. feedback sensitivity to the gas threshold for galaxy formation.


Very Difficult to disentangle baryonic effects in the Simulations!


Some outliers like IC 2574

still unexplainable with Feedback!


4

Alternative Dark Matter Models

Warm Dark Matter (WDM): favored mass range in tension with Ly observation & abundance of high-z galaxies

Self-interacting Dark Matter (SIDM): Needs fine-tuned cross-sections & struggles to explain full range of observations


Still a viable window

No Empirical

Strong constraints


arXiv:2408.00082

Under observational constraints!

Detection so far!

5

with microlensing observations!

Fuzzy Dark Matter

(F(C)DM, BECDM, ULDM, ELBDM, (ultra-light) axion (-like) DM (ULA, ALP))



6


7

FDM potential signature in 21cm power spectrum


Distinct power suppression at high z!


8

FDM mass constraints based on PLin


Need Full Non-

Linear Simulations to get reliable constraints!


9

Governing Equations

  1. Wave Formalism (Schrödinger-Poisson Equations)

    Mean Field Interpretation: Single Macroscopic WF of BEC

    i 2 2

    ℏ∂tψ = − 2m ψ + mVψ

    2V = 4πGm( | ψ |2 − | ψ0 |2 )

    ρ

    m

  2. Madelung Formalism (Fluid Dynamics Representation)


tρ + ∇⋅ (ρv⃗) = 0

1


2 2

ψ = eiS

ρ

tv⃗ + (v⃗ ⋅

)v⃗ = − m

(V

2m )

ρ = m | ψ |2

ρ

= Q

2V = 4πGm(ρ ρ0)

v = m S

Q ill-defined at ρ = 0 !

“Quantum Pressure”

10

Fluid Solver unable to capture interference effects!

Fuzzy Dark Matter Simulations


Stick to SP-Equations for evolution!

11 A. Kunkel et.al. 2024

Challenges in Simulating Fuzzy Dark Matter


Both Mpc-scale and kpc-scales need to be Resolved for accurate evolution


Time step scaling: Δt ∼ Δx2


Hydrodynamical codes are used in N-body Simulation (but Fluid Formulation

For FDM evolution?)


So far sims. restricted to small box sizes of 10Mpc/h


12 Schive, Chieuh, & Broadhurst (2014)

Challenges in Simulating Fuzzy Dark Matter


Both Mpc-scale and kpc-scales need to be Resolved for accurate evolution

Time step scaling: Δt ∼ Δx2

Hydrodynamical codes are used in N-body Simulation (but Fluid Formulation

For FDM evolution?)


So far sims. restricted to small box sizes of 10Mpc/h


13 Schive, Chieuh, & Broadhurst (2014)

Challenges in Simulating Fuzzy Dark Matter


Both Mpc-scale and kpc-scales need to be Resolved for accurate evolution

Time step scaling: Δt ∼ Δx2

Hydrodynamical codes are used in N-body Simulation (but Fluid Formulation

For FDM evolution?)


So far sims. restricted to small box sizes of 10Mpc/h


14 Schive, Chieuh, & Broadhurst (2014)

Physics Informed Neural Networks


General Framework:

𝒟[NN(X, θ); λ] = f(X), X ∈ Ω

ℬ[NN(X, θ); ] = g(X) X ∈ ∂Ω


Custom Loss Function: with PDE and boundary conditions as additional constraints

Adapted from F. Pioch et.al.2023


Pretty Successful in Fluid and Climate Simulations!

Raissi, Yazdani, Karinadakis 2020


15

Outline of the approaches


01

Coordinate based approach

Directly solve the Schrödinger–Poisson (SP) equations using Physics-Informed Neural Networks (PINNs).


02

Generative Modeling

Learn the mapping from Cold Dark Matter (CDM) to Fuzzy Dark Matter (FDM) cosmological simulations at a fixed redshift.


03

Physics-Informed Generative Modeling

Interpolate and simulate FDM cosmological boxes across multiple redshifts using physics-constrained generative models.


16

Schrödinger-Poisson Equations used

λ =

1

potential

λ : the strength of

m i Ψ(x, t) = − λ 2 + 1 V[Ψ(x, t)] Ψ(x, t)

t ( 2 λ )

2V[Ψ(x, t)] = ( |Ψ(x, t) |2 − 1)


λ → 0, Gravitational Potential Term is dominant in the SP Equations!


λ → ∞, Gravitational Potential Term vanishes, Free Schrodinger Equation

representing diffusion!


λ = 1 throughout this work!


17

x

y

Re(ψ)

Schrödinger-Poisson Informed Neural Networks (SPINN)


iℏ∂ ψ = −

2

t

2m

2ψ + mVψ

2V = 4πGm( | ψ |2 − | ψ0 |2 )


z

t


Neural network

Im(ψ)

V

18

x

y

Re(ψ)

Schrödinger-Poisson Informed Neural Networks (SPINN)


iℏ∂ ψ = −

2

t

2m

2ψ + mVψ

2V = 4πGm( | ψ |2 − | ψ0 |2 )


z

t


Neural network

Im(ψ)

V


Use automatic differentiation to calculate derivatives

tψ, 2ψ, ∇2V, . . .

19

x

y

Re(ψ)


Use automatic differentiation to calculate derivatives

tψ, 2ψ, ∇2V, . . .

Schrödinger-Poisson Informed Neural Networks (SPINN)


iℏ∂ ψ = −

2

t

2m

2ψ + mVψ

2V = 4πGm( | ψ |2 − | ψ0 |2 )


Construct physics-informed loss:

L = | (iψ/∂t + ∇2ψ/2 − mVψ) |2 + | ( ∇2V − | ψ2 | + 1) |2


z

t


Neural network

Im(ψ)

V

20

x

y

Re(ψ)


Use automatic differentiation to calculate derivatives

tψ, 2ψ, ∇2V, . . .

Use backpropagation to find network parameters that minimize these losses

Schrödinger-Poisson Informed Neural Networks (SPINN)



Loss for initial and boundary conditions


Construct physics-informed loss:

L = | (iψ/∂t + ∇2ψ/2 − mVψ) |2 + | ( ∇2V − | ψ2 | + 1) |2


z

t


Neural network

Im(ψ)

V

21


Results

Density Predictions

1D NN

1D Sim


Mishra & Tolley, ApJ 2025


3D NN

3D Sim

Unsupervised neural network accurately predicting FDM dynamics using just physics constraints and initial conditions


23

Extrapolation



Evidence of SPINNs' ability to generalize and predict beyond trained time intervals !

24

What about cosmological FDM simulations?

Approaches


Learning the CDM to FDM density mapping

ρCDM(x) → ρFDM(x)

(Essence: Learn the small scale corrections with ML — where FDM diverges — while taking advantage of its numerical equivalence to CDM at large scales.)


Use physics-informed Generative models to simulate/interpolate FDM boxes across redshifts


25

How do diffusion models work?


source: arXiv:1503.03585, 2015


Source: medium.com – Vadim Titko 26

Data for CDM to FDM Mapping I


Courtesy of Simon May


27

Data for CDM to FDM Mapping II


For both the boxes, we extract a sub-volume spanning 0 to 3 Mpc/h on each side.


This 3 Mpc region is then:

Downsampled to a resolution of 576³ voxels.

Further chunked into smaller 64³ voxel boxes to fit within GPU memory constraints.

In total 729 pairs of CDM and FDM boxes!


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Current Hybrid Approach-WGAN (Ongoing work)


Real (FDM)



Generative Model (WGAN)


Input (CDM)

Generated (FDM)


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Fringes not fully resolved!

Current Hybrid Approach-WGANs (Ongoing work)



Real


Real (FDM)

30

Generated (FDM)

Much better!

Current Hybrid Approach-DDPMs(Ongoing work)



Real


Real (FDM)

31

Can be used as an effective compression algorithm for FDM


Training FDM Data size 130 GB, CDM Data 0.8 GB, DDPM Model 1.3 GB



Input (CDM)


Generative Model (DDPM)


Generated (FDM)


130GB

Compressed to

2GB

(FDM) (CDM + DDPM Model)


65 fold compression!

32

What about generalizing to unseen regions? Can these models hallucinate realistically?

Testing the model trained on 0-3 Mpc sided volume beyond the trained spatial region



Actual: 3-6 Mpc volume


DDPM @ test

33

What about generalizing to unseen regions? Can these models hallucinate realistically?

When trained with larger volume cut-outs and some additional losses!



Overlapping patches of 1.5 Mpc

Overlapping patches of 1.5 Mpc

+ Powerspectrum loss

34

Can these models hallucinate realistically?

May be with

Testing the model trained on 0-3 Mpc sided volume beyond the trained spatial region


enough samples!?

Actual: 3-6 Mpc volume

DDPM @ test

35

Data for Physics-Informed GANs

We have 2 Mpc/h sized FDM boxes at redshifts z = {127, 63, 31, 15, 7, 6, 5, 4, 3, 2, 1, 0}.


Out of which z = {7, 6, 5, 4, 3, 2} boxes are chosen for training and z = {1, 0} for test.


Each of these boxes’ original resolution is

24003


This chosen subset of samples is then:

Downsampled to a resolution of 64³ voxels.

36

(Physics-Informed) Generative Models (in progress)

After 8 hrs of training on single GPU


Real


Gen


Yet to include full physics!

2Y |pred = ∇2Y |actual


37

Work in Progress!

(Still to include physics constraints rigorously)

  1. Neural Operators for CDM to FDM mapping


  2. Physics-informed neural operators for painting in FDM small-scale

    features


  3. Supervised PINNs using large-scale CDM simulations as additional data

    constraint


  4. Reproducing Core-Halo Relations for FDM with PINNs

38


THANK YOU!


Question?


Ashutosh Kumar Mishra Email: ashutosh.mishra@epfl.ch


Arxiv Link

39


Back-up

Generative Adversarial Networks (GANs)


Real Images

High Dimensional Sample Space


Generator

G


Discriminator

D

Generated Images

Real


Low Dimensional Latent Space


Fake


41

Fuzzy Dark Matter

Linear theory predicts sharp cutoff in power spectrum due to quantum pressure



42

Direct probe of DM distribution

Power spectrum, halo mass profile


Bechtol et al. 2023


43

Numerical Method (Mocz et. al. 2017)

2nd Order Unitary Spectral Method

( k2 (

0 ))

ψ exp[−i(m/ℏ)(Δt/2)V ]ψ


V IFFT


( k2 (

0 ))

ψ exp[−i(m/ℏ)(Δt/2)V ]ψ

Kick